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xxx-x-10=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -10$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -10$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -10$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -10$$
Respuesta rápida
[src]
______________ / ______________ \
/ ______ | / ______ |
/ \/ 2022 | ___ / \/ 2022 |
3 / 5 + -------- | \/ 3 *3 / 5 + -------- ___ |
\/ 9 1 | \/ 9 \/ 3 |
x1 = - ------------------- - --------------------- + I*|- ------------------------- + ---------------------|
2 ______________ | 2 ______________|
/ ______ | / ______ |
/ \/ 2022 | / \/ 2022 |
6*3 / 5 + -------- | 6*3 / 5 + -------- |
\/ 9 \ \/ 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}\right)$$
______________ / ______________ \
/ ______ | / ______ |
/ \/ 2022 | ___ / \/ 2022 |
3 / 5 + -------- |\/ 3 *3 / 5 + -------- ___ |
\/ 9 1 | \/ 9 \/ 3 |
x2 = - ------------------- - --------------------- + I*|------------------------- - ---------------------|
2 ______________ | 2 ______________|
/ ______ | / ______ |
/ \/ 2022 | / \/ 2022 |
6*3 / 5 + -------- | 6*3 / 5 + -------- |
\/ 9 \ \/ 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2}\right)$$
______________
/ ______
/ \/ 2022 1
x3 = 3 / 5 + -------- + ---------------------
\/ 9 ______________
/ ______
/ \/ 2022
3*3 / 5 + --------
\/ 9
$$x_{3} = \frac{1}{3 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}$$
x3 = 1/(3*(sqrt(2022)/9 + 5)^(1/3)) + (sqrt(2022)/9 + 5)^(1/3)
Suma y producto de raíces
[src]
suma
______________ / ______________ \ ______________ / ______________ \
/ ______ | / ______ | / ______ | / ______ |
/ \/ 2022 | ___ / \/ 2022 | / \/ 2022 | ___ / \/ 2022 | ______________
3 / 5 + -------- | \/ 3 *3 / 5 + -------- ___ | 3 / 5 + -------- |\/ 3 *3 / 5 + -------- ___ | / ______
\/ 9 1 | \/ 9 \/ 3 | \/ 9 1 | \/ 9 \/ 3 | / \/ 2022 1
- ------------------- - --------------------- + I*|- ------------------------- + ---------------------| + - ------------------- - --------------------- + I*|------------------------- - ---------------------| + 3 / 5 + -------- + ---------------------
2 ______________ | 2 ______________| 2 ______________ | 2 ______________| \/ 9 ______________
/ ______ | / ______ | / ______ | / ______ | / ______
/ \/ 2022 | / \/ 2022 | / \/ 2022 | / \/ 2022 | / \/ 2022
6*3 / 5 + -------- | 6*3 / 5 + -------- | 6*3 / 5 + -------- | 6*3 / 5 + -------- | 3*3 / 5 + --------
\/ 9 \ \/ 9 / \/ 9 \ \/ 9 / \/ 9
$$\left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2}\right)\right)\right)$$
=
/ ______________ \ / ______________ \ | / ______ | | / ______ | | ___ / \/ 2022 | | ___ / \/ 2022 | |\/ 3 *3 / 5 + -------- ___ | | \/ 3 *3 / 5 + -------- ___ | | \/ 9 \/ 3 | | \/ 9 \/ 3 | I*|------------------------- - ---------------------| + I*|- ------------------------- + ---------------------| | 2 ______________| | 2 ______________| | / ______ | | / ______ | | / \/ 2022 | | / \/ 2022 | | 6*3 / 5 + -------- | | 6*3 / 5 + -------- | \ \/ 9 / \ \/ 9 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}\right) + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2}\right)$$
producto
/ ______________ / ______________ \\ / ______________ / ______________ \\ | / ______ | / ______ || | / ______ | / ______ || | / \/ 2022 | ___ / \/ 2022 || | / \/ 2022 | ___ / \/ 2022 || / ______________ \ | 3 / 5 + -------- | \/ 3 *3 / 5 + -------- ___ || | 3 / 5 + -------- |\/ 3 *3 / 5 + -------- ___ || | / ______ | | \/ 9 1 | \/ 9 \/ 3 || | \/ 9 1 | \/ 9 \/ 3 || | / \/ 2022 1 | |- ------------------- - --------------------- + I*|- ------------------------- + ---------------------||*|- ------------------- - --------------------- + I*|------------------------- - ---------------------||*|3 / 5 + -------- + ---------------------| | 2 ______________ | 2 ______________|| | 2 ______________ | 2 ______________|| |\/ 9 ______________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / \/ 2022 | / \/ 2022 || | / \/ 2022 | / \/ 2022 || | / \/ 2022 | | 6*3 / 5 + -------- | 6*3 / 5 + -------- || | 6*3 / 5 + -------- | 6*3 / 5 + -------- || | 3*3 / 5 + -------- | \ \/ 9 \ \/ 9 // \ \/ 9 \ \/ 9 // \ \/ 9 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}}\right)\right) \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}} + \sqrt[3]{\frac{\sqrt{2022}}{9} + 5}\right)$$
=
10
$$10$$
10
Respuesta numérica
[src]
x1 = -1.15445365988255 - 1.73155703297599*i
x2 = -1.15445365988255 + 1.73155703297599*i
x3 = 2.30890731976509
x3 = 2.30890731976509